MMJ1133 – FATIGUE AND FRACTURE MECHANICS
E – ENGINEERING FRACTURE MECHANICS
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
E – ENGINEERING FRACTURE MECHANICS
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
WWII: Liberty ships
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
Reprinted w/ permission from R.W. Hertzberg, "Deformation and Fracture
Mechanics of Engineering Materials", (4th ed.) Fig. 7.1(b), p. 262, John Wiley
and Sons, Inc., 1996. (Orig. source: Earl R. Parker, "Behavior of Engineering
Structures", Nat. Acad. Sci., Nat. Res. Council, John Wiley and Sons, Inc., NY,
1957.)
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Course Content:
A - INTRODUCTION
Mechanical failure modes; Review of load and stress analysis –
equilibrium equations, complex stresses, stress transformation,
Mohr’s circle, stress-strain relations, stress concentration; Fatigue
design methods; Design strategies; Design criteria.
B – MATERIALS ASPECTS OF FATIGUE AND FRACTURE
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 3
Static fracture process; Fatigue fracture surfaces; Macroscopic features; Fracture mechanisms; Microscopic features.
C – FATIGUE: STRESS-LIFE APPROACH
Fatigue loading; Fatigue testing; S-N curve; Fatigue limit; Mean
stress effects; Factors affecting S-N behavior – microstructure, size
effect, surface finish, frequency.
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
D – FATIGUE: STRAIN-LIFE APPROACH
Stress-strain diagram; Strain-controlled test methods; Cyclic
stress-strain behavior; Strain-based approach to life estimation;
Strain-life fatigue properties; Mean stress effects; Effects of surface
finish.
E – LINEAR ELASTIC FRACTURE MECHANICS
Fundamentals of LEFM – loading modes, stress intensity factor, K;
Geometry correction factors; Superposition for Mode I; Crack-tip
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 4
Geometry correction factors; Superposition for Mode I; Crack-tip
plasticity; Fracture toughness, KIC ; Plane stress versus plane strain
fracture; Extension to elastic-plastic fracture.
F – FATIGUE CRACK PROPAGATION
Fatigue crack growth; Paris Law; da/dN-∆K; Crack growth test method; Threshold ∆Kth ; Mean stress effects; Crack growth life
integration.
.
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
A branch of mechanics that studies the relationships
between external loads applied to a deformable
body and the intensity of internal forces acting
within the body.
Mechanics of Materials
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
within the body.
The mechanics that describes the response of
materials to loading in the presence of crack or
crack-like defects.
Fracture Mechanics
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
• Introduction
Historical Review
Fracture mechanics approaches
• Linear Elastic Fracture Mechanics
Elastic stress field approach
Crack tip plasticity
Energy balance approach
LEFM testingA Short Course in
Fracture Mechanics
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
LEFM testing
• Elastic-Plastic Fracture Mechanics
J-integral
COD approach
• Fracture Mechanics Concept for Crack Growth
Fatigue crack growth
Dynamic crack growth and arrest
• Time-Dependent Fracture
• Fracture Mechanisms in Metals and Nonmetals
Fracture Mechanics
(Typical course content)
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
References:
• Anderson, T.L., Fracture Mechanics – Fundamentals and Applications, 3rd
edition, CRC Press, FL, USA, 2005.
• Broek, D., Elementary Engineering Fracture Mechanics, Kluwer Academic
Publishers, 1991.
•Atkins, A.G. and Mai, Y.W., Elastic and Plastic Fracture – Metals, Polymers,
ceramics, Composites, Biological Materials, Ellis Horwood Ltd., UK, 1985.
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 7
ceramics, Composites, Biological Materials, Ellis Horwood Ltd., UK, 1985.
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Linear Elastic Fracture Mechanics (LEFM)
� Fracture mechanics within the confines of the theory of
linear elasticity.
� Analytical procedure that relates the stress magnitude and
distribution in the neighborhood of a crack to:
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
� the nominal applied stress
� crack geometry (size, shape) and orientation
� material properties
� An underlying principle is that unstable fracture occurs
when the stress-intensity factor at the crack tip reaches a
critical value.
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Scope of fracture mechanics
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Basic loading of cracked bodies
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Stress field ahead of crack tip
−
=2
3sin
2sin1
2cos
2
θθθσσ
r
ax
+
=2
3sin
2sin1
2cos
2
θθθσσ
r
ay
θθθ
rθ
Westergaard solution
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
=2
3cos
2sin
2cos
2
θθθστ
r
axy
KI is called stress
intensity factor (SIF)
( ) ( )termsorderhigherfr
Kij
Iij += θ
πσ
2
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Stress intensity factor
The stress intensity factor,
KI describes the crack tip
stresses.τxy
σyy
σ
σ
( )θπ
σ ijI
ijr
fr
K
2lim
0
=→
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
2a
Crack
σxy
θr
σ
β- dimensionless parameter
KI has dimension of MPa√m
aK I βσ=
aK I πσ=
For infinite cracked plate
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Stress field at notch tip
Compact tension C(T) specimen
σyy
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
von Mises
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Crack-tip stress
r
K
r
K
Iy
Iy
πσ
θθθπ
σ
2
2
3sin
2sin1
2cos
2
=
+
=
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Crack-tip plasticity
*2 p
IYS
r
K
πσ =
2
2
2
2
*
22
Ip
aKr
σσ
σπ==
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
2222 YSYS
pr σσπ==
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Shape of plastic zone
( )
++
= θθ
σπθ 2
2
sin2
3cos1
4
1
YS
Iy
Kr
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 16
( ) ( ) ( )
++−
= θθν
σπθ 22
2
sin2
3cos121
4
1
YS
Iy
Kr
Plane strain
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
aYK I πσ=
Meaningful parameters are σ and a
Finite width correction
aK I πσ= Infinite cracked plate
σ
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 17
I
w
aY
πsec=
σ
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Finite width correction for SIF
For 2a<<W,
aK I πσ=
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Condition for fracture
Fracture occurs when the applied stress intensity factor, KI
reaches the value of the fracture toughness, KIC of the material
ICI KK ≥
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
σ
aa
aK I πσ1.1=
ICc Ka =πσ1.1
ICc Ka =πσ1.1
At fracture:
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Graphite/ Ceramics/ Semicond
Metals/ Alloys
Composites/ fibers
Polymers
0.5
) Mg alloys
Al alloys
Ti alloys
Steels
20
30
C-C(|| fibers)1
40
506070
100
Al/Al oxide(sf)2
Fracture toughness
represents the resistance of
materials to resist cracking.
Fracture Toughness
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
5
KIc
(MP
a ·
m0
.5
1Si crystal
Glass-soda
Concrete
Si carbide
PC
Glass6
0.5
0.7
2
4
3
10
<100>
<111>
Diamond
PVC
PP
Polyester
PS
PET
0.6
67
Al oxideSi nitride
C/C( fibers)1
Al/Al oxide(sf)2
Al oxid/SiC(w)3
Al oxid/ZrO2(p)4
Si nitr/SiC(w)5
Glass/SiC(w)6
Y2O3/ZrO2(p)4
Based on data in Table B5,
Callister 6e.
materials to resist cracking.
Fracture toughness values
are determined from
fracture toughness tests.
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Effect of thickness
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
Ref. : T.L Anderson, CRC Press, 2005
Plane
stress
Plane
strain
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Plane Stress versus Plane Strain
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Plane stress versus plane strain fracture
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 25
Ref. : Atkins and Mai, 1985
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Stress triaxiality at crack-tip
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 26
Ref. : T.L Anderson, CRC Press, 2005
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Residual strength
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
K-controlled fracture• KI characterizes crack-tip
condition even though the 1/√ r
singularity does not apply to the
plastic zone.
• LEFM ceased to be valid when
the plastic zone size becomes
large relative to key dimensions
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 28
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Case I - Maximum flaw size dictates the design
stress.
σdesign <Kc
Y πamax
σ
fracture
Design considerations
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM
amaxno fracture
fracture
Case II - Design stress dictates the tolerable maximum
flaw size.
amax <1
πKc
Yσdesign
2
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Energy release rate
Change in energy, dU due to
crack growth from a to a+da is
represented by the shaded
area.
drudU
da
∫= σ
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 30
drudU yy∫=0
σ
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Energy release rate
drudU y
da
y∫=0
σ
r
ay
2σσ =
( ) ( )rdaaE
ur −−= 2122 σ
ν
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 31
( )daE
adU 2
2
1 νπσ
−=
( )22
1 νπσ
−=E
a
da
dU
Energy release rate, G
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Energy release rate
( )22
1 νπσ
−==E
a
da
dUG Plane strain
E
aG
πσ 2
= Plane stress
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 32
( )22
1 ν−=E
KG
E
KG
2
=
Plane strain
Plane stress
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Fracture Toughness Test
ASTM E399 Standard Test Method for Plane Strain Fracture Toughness of
Metallic Materials
Sample geometry
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 33
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Fracture Toughness Test
Direction of cut
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 34
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Fracture Toughness Test
Fatigue pre-cracking
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 35
∆=∆
W
af
WB
PK
MMJ1133 – FATIGUE AND FRACTURE MECHANICS
Fracture Toughness Test
Load-gage displacement
=W
af
WB
PK
Q
Q
a
Validity requirements
for KIC
FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 36
Q
YS
Q
PP
KaB
W
a
10.1
5.2,
55.045.0
max
2
≤
≥
≤≤
σ